Common operations in linear algebra

Cross product

Definition

$$ \vec{a} \times \vec{b} = \vvv{a_1}{a_2}{a_3} \times \vvv{b_1}{b_2}{b_3} = \vvv{a_2b_3-a_3b_2}{a_3b_1-a_1b_3}{a_1b_2-a_2b_1} $$

Geometric Interpretation

The absolute value of $\ \vec{a}\times\vec{b}\ $ is equal to the area of the parallelogram that the vectors $\vec{a}$ and $\vec{b}$ span: $$ |\vec{a}\times\vec{b}| = |\vec{a}||\vec{b}|\sin\theta $$

Scalar triple product

Definition

$$ [\vec{a},\vec{b},\vec{c}] = (\vec{a}\times\vec{b})\cdot\vec{c} $$

Alternate notations

$$ [\vec{a},\vec{b},\vec{c}] = \langle\vec{a},\vec{b},\vec{c}\rangle = |\vec{a}\vec{b}\vec{c}| $$

Geometric Interpretation

Geometrically, the scalar triple product $\ (\vec{a}{\small\times}\vec{b}){\small\cdot}\vec{c}\ $ is the (signed) volume of the parallelepiped defined by the three vectors given (Wikipedia).

Properties

Circular shift

The scalar triple product is unchanged under a circular shift of its three operands $(\vec{a},\vec{b},\vec{c})$: $$ [\vec{a},\vec{b},\vec{c}] = [\vec{b},\vec{c},\vec{a}] = [\vec{c},\vec{a},\vec{b}] $$

Determinant

The scalar triple product is equal to the determinant of the $3{\small\times}3$ matrix that has the three vectors either as its rows or its columns (the transposed matrix has the same determinant): $$ [\vec{a},\vec{b},\vec{c}] = det\begin{bmatrix}a_1&a_2&a_3\\b_1&b_2&b_3\\c_1&c_2&c_3\end{bmatrix} = det(\vec{a},\vec{b},\vec{c}) $$